3.4.32 \(\int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx\)

Optimal. Leaf size=28 \[ \frac {\sqrt {x^6+1}}{3}+\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {446, 80, 63, 207} \begin {gather*} \frac {\sqrt {x^6+1}}{3}+\frac {1}{3} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/3 + ArcTanh[Sqrt[1 + x^6]]/3

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {-1+x^6}{x \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=\frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \left (\sqrt {x^6+1}+\tanh ^{-1}\left (\sqrt {x^6+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]

[Out]

(Sqrt[1 + x^6] + ArcTanh[Sqrt[1 + x^6]])/3

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IntegrateAlgebraic [A]  time = 0.04, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6}}{3}+\frac {1}{3} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/3 + ArcTanh[Sqrt[1 + x^6]]/3

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fricas [A]  time = 0.48, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

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giac [A]  time = 0.29, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

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maple [A]  time = 0.22, size = 27, normalized size = 0.96

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}}{3}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{3}\)	\(27\)
default	\(\frac {\sqrt {x^{6}+1}}{3}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{3}\)	\(29\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{6 \sqrt {\pi }}-\frac {\left (-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)/x/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(x^6+1)^(1/2)+1/3*ln(((x^6+1)^(1/2)+1)/x^3)

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maxima [A]  time = 0.55, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} + 1} + \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)/x/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 1) + 1/6*log(sqrt(x^6 + 1) + 1) - 1/6*log(sqrt(x^6 + 1) - 1)

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mupad [B]  time = 0.30, size = 20, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{3}+\frac {\sqrt {x^6+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)/(x*(x^6 + 1)^(1/2)),x)

[Out]

atanh((x^6 + 1)^(1/2))/3 + (x^6 + 1)^(1/2)/3

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sympy [A]  time = 28.44, size = 39, normalized size = 1.39 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{3} - \frac {\log {\left (-1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{6} + \frac {\log {\left (1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)/x/(x**6+1)**(1/2),x)

[Out]

sqrt(x**6 + 1)/3 - log(-1 + 1/sqrt(x**6 + 1))/6 + log(1 + 1/sqrt(x**6 + 1))/6

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